Inverse scattering for Schrödinger operators with Miura potentials: I. Unique Riccati representatives and ZS-AKNS systems

C. Frayer, R. O. Hryniv, Ya V. Mykytyuk, P. A. Perry

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21 Scopus citations

Abstract

This is the first in a series of papers on scattering theory for one-dimensional Schrödinger operators with highly singular potentials q ε H-1loc(ℝ) . In this paper, we study Miura potentials q associated with positive Schrödinger operators that admit a Riccati representation q = u′ + u2 for a unique u ε L 1 (ℝ) ∩ L2 (ℝ). Such potentials have a well-defined reflection coefficient r(k) that satisfies |r(k)| < 1 and determines u uniquely. We show that the scattering map S : u → r is real analytic with real-analytic inverse. To do so, we exploit a natural complexification of the scattering map associated with the ZS-AKNS system. In subsequent papers, we will consider larger classes of potentials including singular potentials with bound states.

Original languageEnglish
Article number115007
JournalInverse Problems
Volume25
Issue number11
DOIs
StatePublished - 2009

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Signal Processing
  • Mathematical Physics
  • Computer Science Applications
  • Applied Mathematics

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